Conservative and Semiconservative Random Walks: Recurrence and Transience

Abstract

In the present paper we define conservative and semiconservative random walks in $\mathbb{Z}^d$ and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (P\'olya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in $\mathbb{Z}^d$ by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in $\mathbb{Z}^d$ for any $d\geq3$ and transient random walks in $\mathbb{Z}^2$.